Let f: X → Y be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on a topological space. The direct image functor
sends a sheaf F on X to its direct image presheaf f∗F on Y, defined on open subsets U of Y by
This turns out to be a sheaf on Y, and is called the direct image sheaf or pushforward sheaf of F along f.
Since a morphism of sheaves φ: F → G on X gives rise to a morphism of sheaves f∗(φ): f∗(F) → f∗(G) on Y in an obvious way, we indeed have that f∗ is a functor.
Example
If Y is a point, and f: X → Y the unique continuous map, then Sh(Y) is the category Ab of abelian groups, and the direct image functor f∗: Sh(X) → Ab equals the global sections functor.
Variants
If dealing with sheaves of sets instead of sheaves of abelian groups, the same definition applies. Similarly, if f: (X, OX) → (Y, OY) is a morphism of ringed spaces, we obtain a direct image functor f∗: Sh(X,OX) → Sh(Y,OY) from the category of sheaves of OX-modules to the category of sheaves of OY-modules. Moreover, if f is now a morphism of quasi-compact and quasi-separated schemes, then f∗ preserves the property of being quasi-coherent, so we obtain the direct image functor between categories of quasi-coherent sheaves.[1]
A similar definition applies to sheaves on topoi, such as étale sheaves. There, instead of the above preimage f−1(U), one uses the fiber product of U and X over Y.
The direct image functor is left exact, but usually not right exact. Hence one can consider the right derived functors of the direct image. They are called higher direct images and denoted Rq f∗.
One can show that there is a similar expression as above for higher direct images: for a sheaf F on X, the sheaf Rq f∗(F) is the sheaf associated to the presheaf
- ,
where Hq denotes sheaf cohomology.
In the context of algebraic geometry and a morphism of quasi-compact and quasi-separated schemes, one likewise has the right derived functor
as a functor between the (unbounded) derived categories of quasi-coherent sheaves. In this situation, always admits a right adjoint .[2] This is closely related, but not generally equivalent to, the exceptional inverse image functor , unless is also proper.